Multiple-invariant space-variant optical processing

ABSTRACT

A multiple-invariant, space variant optical processor in which two functions described by any number of separate distortion parameters can be correlated with no loss in signal-to-noise ratio of the correlation. The unknown distortion parameters can also be determined in this scheme. Experimental confirmation of the key step, determination of the non-linear phase portion of a complex optical transform can be provided.

STATEMENT OF GOVERNMENT INTEREST

The invention described herein may be manufactured and used by or forthe Government for governmental purposes without the payment of anyroyalty thereon.

BACKGROUND OF THE INVENTION

This invention relates to a multiple-invariant, space-variant opticalprocessor or correlator and, more particularly, the invention isconcerned with providing a multiple-invariant space-variant opticalprocessor in which two functions, described by any number of separatedistortion parameters, can be correlated with no loss in signal-to-noiseratio of the correlation. The unknown distortion parameters can also bedetermined in this scheme wherein the key step includes thedetermination of the non-linear phase portion of a complex opticaltransform.

The parallel processing and real-time features of optical processorshave not been sufficient alone for these systems to see extensive use inpattern recognition. One of the reasons for this lack of practical usefor optical correlators has been their lack of flexibility and thelimited number of operations achievable in these processors. Hybridoptical/digital processors and space variant optical processors haveincreased the flexibility of these systems and included non-linearoperations and space variant systems to the repertoire of opticalprocessors. Recently, a space-variant, distortion-invariant opticalcorrelator using coordinate transformations that allows correlation oftwo functions that are distorted versions of one another has beenreported on (D. Casasent and D. Psaltis, Proc. IEEE, 65, 77 of 1977).However, this approach could only be applied to functions that weredistorted by at the most, two separate distortions (for atwo-dimensional system). Since more than two distortions occur inpractice in image pattern recognition problems, optical processors mustaddress such real issues if they are to be viable candidate patternrecognition systems. In the hereinafter described processor, theformulation of a multiple-invariant, space-variant optical processorthat is invariant to multiple distortions is considered. A generaltheoretical analysis is presented first, followed by several specificcases and implementation methods.

Since these multiple-invariant systems will be formulated using spacevariant processing methods, specifically by use of coordinatetransformations, the general formulation of a distortion-invariant,space variant processor is reviewed. A one-dimensional example isconsidered for simplicity. Let f(x) be the original undistortedfunction. The distorted function f'(x) is described by

    f'(x)=f(x')=f[g(x,a)],                                     (1)

where g(x,a) is the distorting function and "a" is the unknowndistortion parameter. To realize a space variant processor (invariant tothe g(x,a) distorting function), a coordinate transformation ξ=h⁻¹ (x)is applied to both functions. This produces two new functions f₁ (ξ) andf'₁ (ξ), which can be used in a conventional space invariant correlatorto achieve correlation invariant to the distortion g(x,a).

The coordinate transformation is chosen to convert the distortion into ashift ξ₀ for all values of "a." Given g(x,a), the coordinatetransformation can be found from: ##EQU1## These new coordinatetransformed functions f₁ (x) and f'₁ (x) are shifted versions of oneanother. Thus a conventional space invariant system can be used tocorrelate them. The intensity of the output correlation peak will beindependent of "a" and the location of the output correlation peak willbe proportional to "a". These basic principles are used in severalstages in the present multiple invariant system.

Prior space variant processors using coordinate transformations canaccommodate only one distortion parameter per axis (one distortionparameter for a 1-D system or two distortions for a 2-D system, or ingeneral one distortion parameter per axis). If the distorting functiondepends on more than one parameter, then the distorted function is:

    f'(x)=f(x')=f[g(x,a.sub.1,a.sub.2)],                       (3)

where 1-D functions are used and the distortion function g is describedby two distortion parameters a₁ and a₂. Then from Eq. (2), we see thatsince h⁻¹ (x) depends on ∂g/∂a_(i) and since

    .sup.∂ g/.sup.∂ a.sub.i ≠.sup.∂ g/.sup.∂ a.sub.j  ( 4)

each of the distortion parameters requires a different coordinatetransformation h⁻¹ (x). It thus follows that the maximum number ofdistortion parameters that coordinate transformation processing oneaccommodate equals the number of dimensions of the processing system. Inthis context, a distortion is "multiple" if it is described by moreparameters than coordinate transformation processing can accommodate.

When the distorted function is described by Eq. (3), we can achievemultiple invariance by: (a) scanning through all values of theadditional distorting parameter, or (b) elimination of the additionaldistorting parameter by filtering in the transform plane.

The first approach involves construction of a number of systemsinvariant to one of the parameters (say a₁), with each systemcorresponding to one value of the second parameter a₂. If these systemscover all values of a₂, the output of one of them will be the same forall values of a₁ and a₂ for all input functions related by Eq. (3). Thisapproach has been used in multiple holographic matched spatial filtersystems, conventional multichannel Doppler signal processors, and inoptical correlators using mechanical movement of components to effect ascale or rotational search. However, these solutions require a largeprocessing space (if parallel) or are slow (if sequential).

The second approach is more attractive. It uses the fact that a shift inthe input coordinates can be transformed into a linear phase factor inFourier transform space. Elimination of this phase factor alsoeliminates any shifts in the input coordinates. Until now only theformation of the magnitude of the Fourier transform has been suggestedas a method of achieving multiple invariance. We consider thisformulation and its shortcomings in detail below.

With the reference function f(x) and the distorted input function f'(x)defined by Eq. (3), we can realize multiple invariance by first applyingthe transformation x=h(ξ) to f'(x). This transformation must satisfy

    g(x,a.sub.1 a.sub.2)=g[h(ξ-ξ.sub.0),a.sub.2 ],       (5)

where ξ₀ =ξ₀ (a₁) is a constant depending only on the unknown distortionparameter a₁. The transformation x=h(ξ) is determined from Eq. (2). Thisyields a new function:

    f"(ξ-ξ.sub.0,a.sub.2)=f{g[h(ξ-ξ.sub.0),a.sub.2 ]}. (6)

If we attempted to correlate f and f" in a conventional space invariantprocessor, the peak intensity of the correlation will fluctuate as a₂changes. Using a 2-D multichannel processor, each channel can beadjusted to correspond to different a₂ values. This will solve the 1-Dproblem presently being discussed but not the 2-D version of it.

To address the real problem, elimination of the distortion parameter a₁,we form the Fourier transform of f and f". This yields

    [f"(ξ-ξ.sub.0,a.sub.2)]=exp(-jωξ.sub.0)F"(ω), (7)

where F" is the Fourier transform of f" and ω is the radian spatialfrequency variable in the Fourier transform plane. If we form themagnitude of the Fourier transform of f", we will have removed thedependence on a₁. We can then retransform F"(ω) and recover f"independent of a₁. This new f" function and f can then be correlated,except that we have lost the phase of F(ω) and hence potentially usefulinformation about f". The magnitude of the Fourier transform representsthe frequencies present in the input and the phase of the Fouriertransform is affected by the distribution of these frequencies in theinput. Thus some information is lost, since two functions with the samefrequency content but different spatial distributions of thesefrequencies could produce identical results. This could lead to falsecorrelations for such functions. The extent of this as a problem dependsupon the specific functions and application.

However, two processors can be cascaded as in FIG. 1. The firstprocessor consists of a coordinate transformation that converts a₁ to ashift by ξ₀ as before. The magnitude of the Fourier transform of thisfunction (together with f subjected to the same transformation andFourier transform magnitude operation) can then be used as the input toa correlator invariant to a₂ (as a₂ affects the magnitude of the Fouriertransform). The location of the output correlation peak will then beproportional to a₂ as in conventional space variant processing bycoordinate transformations.

f and f' (the original distorted function) are then used as inputs to asecond space variant correlator (invariant to a₁). This secondcorrelator is made adjustable so that it responds to different a₂values. The output of the second correlator then provides a correlationinvariant to a₁ and a₂ and the location of the output correlation peakis proportional to a₁. Thus a₁ and a₂ can be determined and multipledistortion achieved in this system.

This method has several limitations besides the loss of phaseinformation associated with the formation of the magnitude of theFourier transform. First, the coordinate distortion described by a₂ mayor may not be preserved as a coordinate transformation in Fourier space.Furthermore, in the presence of noise, the magnitude of the Fouriertransform is not completely unaffected by a shift since interferenceoccurs between the signal and the noise.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a multiple-invariant, space-variantoptical correlator using the magnitude of the Fourier transform; and

FIG. 2 is a schematic block diagram of a multiple-invariant,space-variant optical correlator with no phase loss of information.

DESCRIPTION OF A PREFERRED EMBODIMENT

The major problem with achieving multiple invariance by forming themagnitude of the Fourier transform of the coordinate transformedfunction was the loss of phase that occurred. The hereinafter disclosedinvention describes a method whereby the phase can be extracted from acomplex wavefront. For this present application, we describe this phaseextraction process as follows.

At the input plane we record f(x-x₀)+f(-x+x₀) and detect the magnitudeof its Fourier transform on a TV camera. After thresholding the outputsignal with a limiter and normalizing the width of all fringes by amonostable in the video line, we obtain a function T(ω) whose derivativeis:

    dT(ω)/dω=[S in(q)][dφ(ω)/dω+x.sub.0 ](8)

where φ(ω) is the phase of the Fourier transform F(ω) of f(x) and q is aconstant.

Recall that the phase of the Fourier transform of the function in themultiple invariant correlator contains a portion that is non-linear in ω(due to the spatial distribution of the function itself) and a portionthat is linear in ω (due to the location of the function in the inputplane). By taking the Fourier transform of Eq. (8), the dc term isproportional to the shift in the input coordinates. This value can thenbe used to properly position the input function. Alternatively, thederivative of Eq. (8) can be formed (this eliminates the constant termx₀ and all linear phase terms in the transform) and integrated twice.This yields the desired phase function free of all linear terms. Thisphase function can then be combined with the magnitude of the Fouriertransform, and inverse transformed to reconstruct the input functionfree from a shift.

The schematic block diagram of a multiple invariant correlator thatavoids the phase loss problems of the prior system is shown in FIG. 2.Recall that the object of this system is to correlate two functions:f'(x) defined by Eq. (3) and the undistorted function f(x) and todetermine the unknown distortion parameters a₁ and a₂. For simplicity,only the f'(x) channel of this system is shown in FIG. 2. Theundistorted reference function f(x) is operated upon similarly.

As shown in FIG. 2, a coordinate transformation x=h₁ (ξ₁) is applied tothe distorted input function f'(x)=f(x'), where from Eq. (3)

    x'=g(x,a.sub.1,a.sub.2).                                   (9)

The coordinate transformation ξ₁ =h⁻¹ ₁ (x) affects only the distortionparameter a₁ and converts its effect to a shift by ξ₀ in the ξ₁coordinate, such that:

    g[h.sub.1 (ξ.sub.1),a.sub.1,a.sub.2 ]=g.sub.1 (ξ.sub.1 -ξ.sub.01,a.sub.2),                                    (10)

where ξ₀₁ =ξ₀₁ (a₁) is a constant dependent on the unknown parameter a₁only. The Fourier transform of

    f.sub.1 (ξ.sub.1 -ξ.sub.01,a.sub.2)=f[g.sub.1 (ξ.sub.1 -ξ.sub.01,a.sub.2)]                                    (11)

is F(ω)exp(jωξ₀₁). Using the linear phase extraction scheme of the firstthree paragraphs under the DESCRIPTION OF A PREFERRED EMBODIMENT, we candetermine ξ₀₁ (a₁) and hence a₁.

If we use the ω₀₁ (a₁) value found from the phase extraction system toshift f₁ (ξ₁ -ξ₀₁,a₂), we obtain f₁ (ξ₁,a). Applying the inversetransform ξ₁ =h₁ ⁻¹ (x) to this function we obtain:

    f{g.sub.1 [h.sub.1.sup.-1 (x),a.sub.2 ]}=f[g(x,a.sub.2)],  (12)

which is now independent of a₁.

We can now apply conventional space variant correlation by coordinatetransformation to this function and f(x). We achieve this by applyingthe coordinate transformation ξ₂ =h₂ ⁻¹ (x) to both f(x) and f[g(x,a₂)],where ξ₂ is selected to satisfy Eq. (2). These two new coordinatetransformed functions f(ξ₂) and f(ξ₂ -ξ₀₂) (where ξ₀₂ =ξ₀₂ (a₂) is aconstant dependent on the unknown a₂ only) are then used as inputs to aconventional space invariant correlator. The output correlation peak isinvariant to the distortion described by Eqs. (3) and (9). The locationof the output correlation peak is proportional to ξ₀₂ and hence to a₂.Thus the desired distortion invariant correlation has been realized andthe unknown distortion parameters a₁ and a₂ determined.

Experimental verification of the results hereinbefore disclosed isdescribed in our article entitled Multiple-Invariant Space-VariantOptical Processors, appearing in Applied Optics, Vol. 17, No. 4, pages655-659 including the references cited therein.

Although the invention has been illustrated in the accompanying drawingand described in the foregoing specification in terms of a preferredembodiment thereof, the invention is not limited to this embodiment. Itwill be apparent to those skilled in the art that certain changesmodifications and substitutions can be made without departing from thetrue spirit and scope of the appended claims.

Having thus set forth the nature of our invention, what we claim anddesire to secure by Letters Patent of the United States is:
 1. Amultiple invariant space variant optical processing method forperforming optical correlations on input signals having known multipledistortions comprising the steps of:(a) performing a particularcoordinate transformation that is related to one of said multipledistortions on the signal having multiple distortions; (b) performing anoptical Fourier transformation on the coordinate transformed signal; (c)extracting the linear part of the phase of the Fourier transformedsignal for said one of said multiple distortions; (d) repeating thesteps of performing a coordinate transformation, performing an opticalFourier transformation and extracting the linear part of the phase ofthe Fourier transformed signal on the resultant signal from theextracting step for each of said known multiple distortions; and (e)correlating the resultant signal, free from said known multipledistortions, with an undistorted reference signal.